Problem: Let $f(x)=5\log_2(x)$. Find $f'(x)$. Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{5\ln(2)}{\ln(x)}$ (Choice B) B $\dfrac{5}{\ln(2)x}$ (Choice C) C $\dfrac{5}{\log_2(x)}$ (Choice D) D $\dfrac{5\ln(2)}{x}$
Solution: The expression for $f(x)$ includes a logarithmic term. Remember that the derivative of the general logarithmic term $\log_a(x)$ (where $a$ is any positive constant and $a\neq 1$ ) is $\dfrac{1}{\ln(a)\cdot x}$. Put another way, $\dfrac{d}{dx}[\log_a(x)]=\dfrac{1}{\ln(a)\cdot x}$. [Is there an easy way to memorize that?] We can use this to find the derivative of the function as shown below. $\begin{aligned} f'(x)&=\dfrac{d}{dx}[5\log_2(x)] \\\\ &=5\dfrac{d}{dx}[\log_2(x)] \\\\ &=5\cdot\dfrac{1}{\ln(2)x} \\\\ &=\dfrac{5}{\ln(2)x} \end{aligned}$ In conclusion, $f'(x)=\dfrac{5}{\ln(2)x}$.